3.446 \(\int x (-\sqrt {1-x}-\sqrt {1+x}) (\sqrt {1-x}+\sqrt {1+x}) \, dx\)

Optimal. Leaf size=21 \[ \frac {2}{3} \left (1-x^2\right )^{3/2}-x^2 \]

[Out]

-x^2+2/3*(-x^2+1)^(3/2)

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Rubi [A]  time = 0.11, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6688, 6742, 261} \[ \frac {2}{3} \left (1-x^2\right )^{3/2}-x^2 \]

Antiderivative was successfully verified.

[In]

Int[x*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

-x^2 + (2*(1 - x^2)^(3/2))/3

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int x \left (-\sqrt {1-x}-\sqrt {1+x}\right ) \left (\sqrt {1-x}+\sqrt {1+x}\right ) \, dx &=-\int x \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx\\ &=-\int \left (2 x+2 x \sqrt {1-x^2}\right ) \, dx\\ &=-x^2-2 \int x \sqrt {1-x^2} \, dx\\ &=-x^2+\frac {2}{3} \left (1-x^2\right )^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 1.00 \[ \frac {2}{3} \left (1-x^2\right )^{3/2}-x^2 \]

Antiderivative was successfully verified.

[In]

Integrate[x*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

-x^2 + (2*(1 - x^2)^(3/2))/3

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fricas [A]  time = 0.43, size = 25, normalized size = 1.19 \[ -x^{2} - \frac {2}{3} \, {\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-(1-x)^(1/2)-(1+x)^(1/2))*((1-x)^(1/2)+(1+x)^(1/2)),x, algorithm="fricas")

[Out]

-x^2 - 2/3*(x^2 - 1)*sqrt(x + 1)*sqrt(-x + 1)

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giac [B]  time = 0.39, size = 54, normalized size = 2.57 \[ -{\left (x + 1\right )}^{2} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + 2 \, x + 2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-(1-x)^(1/2)-(1+x)^(1/2))*((1-x)^(1/2)+(1+x)^(1/2)),x, algorithm="giac")

[Out]

-(x + 1)^2 - 1/3*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) - sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + 2*x + 2

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maple [A]  time = 0.00, size = 26, normalized size = 1.24 \[ -x^{2}-\frac {2 \sqrt {x +1}\, \sqrt {-x +1}\, \left (x^{2}-1\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-(-x+1)^(1/2)-(x+1)^(1/2))*((-x+1)^(1/2)+(x+1)^(1/2)),x)

[Out]

-x^2-2/3*(x+1)^(1/2)*(-x+1)^(1/2)*(x^2-1)

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maxima [A]  time = 1.47, size = 17, normalized size = 0.81 \[ -x^{2} + \frac {2}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-(1-x)^(1/2)-(1+x)^(1/2))*((1-x)^(1/2)+(1+x)^(1/2)),x, algorithm="maxima")

[Out]

-x^2 + 2/3*(-x^2 + 1)^(3/2)

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mupad [B]  time = 3.06, size = 25, normalized size = 1.19 \[ -x^2-\frac {2\,\left (x^2-1\right )\,\sqrt {1-x}\,\sqrt {x+1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x*((x + 1)^(1/2) + (1 - x)^(1/2))^2,x)

[Out]

- x^2 - (2*(x^2 - 1)*(1 - x)^(1/2)*(x + 1)^(1/2))/3

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sympy [A]  time = 101.26, size = 110, normalized size = 5.24 \[ \frac {x^{3}}{3} + x - \frac {\left (x + 1\right )^{3}}{3} + 4 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) - 4 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) + 1 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-(1-x)**(1/2)-(1+x)**(1/2))*((1-x)**(1/2)+(1+x)**(1/2)),x)

[Out]

x**3/3 + x - (x + 1)**3/3 + 4*Piecewise((x*sqrt(1 - x)*sqrt(x + 1)/4 + asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1
) & (x < 1))) - 4*Piecewise((x*sqrt(1 - x)*sqrt(x + 1)/4 - (1 - x)**(3/2)*(x + 1)**(3/2)/6 + asin(sqrt(2)*sqrt
(x + 1)/2)/2, (x >= -1) & (x < 1))) + 1

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